Nonexistence of positive weak solutions of elliptic inequalities

In this paper we give sufficient conditions for the nonexistence of positive entire weak solutions of coercive and anticoercive elliptic inequalities, both of the p$p$-Laplacian and of the mean curvature type, depending also on u$u$ and x$x$ inside the divergence term, while a gradient factor is included on the right-hand side. In particular, to prove our theorems we use a technique developed by Mitidieri and Pohozaev in [E. Mitidieri, S.I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001) 1–362], which relies on the method of test functions without using comparison and maximum principles. Their approach is essentially based first on a priori estimates and on the derivation of an asymptotics for the a priori estimate. Finally nonexistence of a solution is proved by contradiction.